Let \(\lbrace d_n:n\in \mathbb {N}\rbrace \)  be an orthonormal basis of \(L^2(X;\mathcal {H})\) . For each \(n \in \mathbb {N},\)  define a function \(s_n:X\rightarrow \mathbb {R}, x\mapsto s_n(x):=\langle P_{J_{\mathcal {A}}(x)}|_{J_{\mathcal {A}^{\prime }}(x)} d_n, d_n\rangle .\)  By definition of range function, the sequence \(s_n\)  is measurable. Denote \(s: = \sup _{n \in \mathbb {N}} s_n\)  defined by, \(x\mapsto \mathfrak {S}( J_{\mathcal {A}}(x), J_{\mathcal {A}^{\prime }}(x))=\sup _{n \in \mathbb {N}}\langle P_{J_{\mathcal {A}}(x)}|_{J_{\mathcal {A}^{\prime }}(x)} d_n, d_n\rangle =\left\Vert P_{ J_{\mathcal {A}^{\prime }}(x)}|_{ J_{\mathcal {A}}(x)}\right\Vert .\)  Then the map \(s\)  is weakly measurable, where the last equality holds because projection maps are self adjoint. Let \(c=\mathfrak {S}(S_{\mathcal {D}}(\mathcal {A}),S_{\mathcal {D}}(\mathcal {A}^{\prime }))=\Vert P_{S_{\mathcal {D}}(\mathcal {A}^{\prime })}|_{S_{\mathcal {D}}(\mathcal {A})}\Vert \)  and \(\tilde{c}=-sup_{x\in \Omega }\big \lbrace \mathfrak {S}\big (J_{\mathcal {A}}(x), J_{\mathcal {A}^{\prime }}(x)\big )\big \rbrace \) . For \(u\in S_{\mathcal {D}}(\mathcal {A})\) , we have \(\Vert P_{J_{\mathcal {A}^{\prime }}(x)}u(x)\Vert \le \mathfrak {S}(J_{\mathcal {A}}(x), J_{\mathcal {A}^{\prime }}(x))\Vert u(x)\Vert ,\)  for a.e. \(x \in X\)  and hence
\(\Vert P_{S_{\mathcal {D}}(\mathcal {A}^{\prime })}(u)\Vert ^2=\int _{X}\Vert  P_{S_{\mathcal {D}}(\mathcal {A}^{\prime })}(u)(x)\Vert ^2\ {d{\mu _{X}}(x)}&= \int _{X} \Vert P_{J_{\mathcal {A}^{\prime }}(x)}u(x)\Vert ^2\ {d{\mu _{X}}(x)}=\int _{\Omega }\Vert P_{J_{\mathcal {A}^{\prime }}(x)}u(x)\Vert ^2\ {d{\mu _{X}}(x)}\\&\le \tilde{c}^2\int _{X}\Vert u(x)\Vert ^2\ {d{\mu _{X}}(x)}\le \tilde{c}^2 \Vert u\Vert ^2,\) 

by observing \(P_{S_{\mathcal {D}}(\mathcal {A}^{\prime })}(u)(x) = P_{J_{\mathcal {A}^{\prime }}(x)}u(x)\)  given in [1]}. Taking the supremum over all \(u\in S_{\mathcal {D}}(\mathcal {A})\)  we have \(c\le \tilde{c}\) .
By choosing \(c<\tilde{c},\)  there exist \(\epsilon >0\)  and a measurable subset \(\Delta \subseteq \Omega \) , with \(\mu _{X}{(\Delta )}>0\)  such that \(u+\epsilon <\mathfrak {S}(J_{\mathcal {A}}(x),J_{\mathcal {A}^{\prime }}(x))\)  for a.e. \(x\in X\) . Construct \(\Delta _i=\lbrace x\in S_{\mathcal {D}}(\mathcal {A}): 0<(c+\epsilon )\Vert P_{J_{\mathcal {A}}(x)}d_i\Vert \le \Vert  P_{J_{\mathcal {A}^{\prime }}(x)}(P_{J_{\mathcal {A}}(x)}d_i)\Vert \rbrace .\) 
Then \(X=\bigcup _{i=1}^{\infty }\Delta _i\) . Hence there exists \(i_0\)  such that \(\mu _{X}(\Delta _{i_0})>0.\)  Define a function \(h\)  by
\(h(x)= {\left\lbrace \begin{array}{ll}P_{J_{\mathcal {A}}(x)}d_i,\ \mbox{if } x \in \Delta _{i_0},\\0, \ \mbox{otherwise}.\end{array}\right.}\) 
Then, \(h\in S_{\mathcal {D}}(\mathcal {A})\)  and hence the following calculation
\(\Vert P_{S_{\mathcal {D}}(\mathcal {A}^{\prime })}h\Vert ^2&=\int _{\Omega }\Vert P_{J_{\mathcal {A}^{\prime }}(x)}h(x)\Vert ^2\ {d{\mu _{X}}(x)}=\int _{\Delta _{i_0}}\Vert P_{J_{\mathcal {A}^{\prime }}(x)}P_{J_{\mathcal {A}}(x)}d_{i_0}\Vert ^2\ {d{\mu _{X}}(x)}\\&\ge (c+\epsilon )^2 \int _{\Delta _{i_0}}\Vert P_{J_{\mathcal {A}}(x)}d_{i_0}\Vert ^2\ {d{\mu _{X}}(x)}=(c+\epsilon )^2\int _{ \Delta _{i_0}}\Vert h(x)\Vert ^2\ {d{\mu _{X}}(x)}=(c+\epsilon )^2\Vert h\Vert ^2>0,\) 

implies that \(c=\mathfrak {S}(S_{\mathcal {D}}(\mathcal {A}),S_{\mathcal {D}}(\mathcal {A}^{\prime }))\ge \frac{\Vert P_{S_{\mathcal {D}}(\mathcal {A}^{\prime })}h\Vert }{\Vert h\Vert }\ge c+\epsilon \) , a contradiction. So \(c\ge \tilde{c}\) . Hence \(c=\tilde{c}\) .
For the remaining part, first observe that the fibers
\(\lbrace \varphi _i(x) : i \in I\rbrace \) 
and \(\lbrace \psi _i(x) : i \in I\rbrace \)  are frames for \(J_\mathcal {A}(x)\)  and \(J_\mathcal {A}^{\prime }(x)\)  for a.e. \(x \in X,\)  since \(\mathcal {E}_{\mathcal {D}}(\mathcal {A})\)  and \(\mathcal {E}_{\mathcal {D}}(\mathcal {A}^{\prime })\)  are frames for \(\mathcal {S}_\mathcal {D}(\mathcal {A})\)  and \(\mathcal {S}_\mathcal {D}(\mathcal {A}^{\prime })\)  by [2]}. The hypothesis imply that \(T_{\mathcal {A}}(x), T_{\mathcal {A}^{\prime }}(x), T_{\mathcal {A}}^*(x),T_{\mathcal {A}^{\prime }}^*(x), G_{\mathcal {A}}(x), G_{\mathcal {A}^{\prime }}(x)\)  have closed range and their pseudo inverses exist. Now, the result follows using [3]}.
